Multisensor triplet Markov fields and theory of evidence

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62 W. Pieczynski, D. Benboudjema / Image and Vision Computing 24 (2006) 61–69 the prior information and the possibly evidential aspects of the sensors. The first integration can be of interest when the hidden scene is non-stationary, and the second one enables one to extend different classical multisensor images analysis (see [14,29,35], among others). Let us mention that similar extensions have been proposed in the case of Markov chains. In fact, hidden Markov chains (HMC) have been generalized to pairwise Markov chains (PMC [23]), and triplet Markov chains (TMC [25]). In particular, evidential priors have been introduced in HMC [7,28], resulting in an improvement in the efficiency of the unsupervised segmentation of non-stationary chains [12]. Moreover, still more complex models, including partially Markov chains or semi-Markov chains, have been recently proposed in [26,27]. All these models can be possibly extended to Markov trees, and some first ideas are presented in [13]. However, although these very general ideas have inspired the main ideas of the present paper, the Markov field models—and the related problems such as, for instance, parameter estimation—are very different from the Markov chain models and the related problems. Therefore, below we will concentrate on Markov fields with no further consideration of Markov chains. The organization of the paper is as follows. The basic notions and calculations relating to the theory of evidence are recalled in Section 2, while Section 3 is devoted to the introduction of evidential priors in the context of Markov fields. The use of evidential sensors is discussed in Section 4, and the general model, including evidential priors and evidential sensors, is specified in Section 5. Section 6 contains conclusions and some future prospects. 2. Theory of evidence Let us consider a finite set of classes UZ{u 1 ,.u k }, and its power set P(U)Z{A 1 ,.A q }, with qZ2 k . A function M from P(U) to [0,1] is called a ‘basic belief assignment’ (bba) if M(:)Z0 and P A2PðUÞ MðAÞZ1. A bba M defines a ‘plausibility’ function Pl from P(U) to [0,1] by PlðAÞZ P AhBs: MðBÞ, and a ‘credibility’ function Cr from P(U) to [0,1] by CrðAÞZ P B3A MðBÞ. For a given bba M, the corresponding plausibility function Pl and credibility function Cr are linked by Pl(A)CCr(A c )Z1. So, each of them defines the other. Conversely, Pl and Cr can be defined by some axioms, and each of them defines an unique corresponding bba M. More precisely, Cr is a function from P(U) to [0,1] verifying Cr(:)Z0, Cr(U)Z1, and Cr g j2J A j R PIs: I3J ðK1ÞjIjC1 Cr h j2I A j , and Pl is a function from P(U) to [0,1] verifying analogous conditions, with % instead of R in the third one. A credibility function Cr verifying such conditions is also the credibility function defined by the bba MðAÞZ P B3A ðK1Þ jAKBj CrðBÞ. Finally, each of the three functions M, Pl, and Cr can be defined in an axiomatic way, and each of them defines the two others. Furthermore, when M is null outside singletons, the corresponding Pl and Cr are equal and become a classical probability. Thus a probability is obtained for a particular M, which will be called ‘probabilistic’ in the following. When two bbas M 1 , M 2 represent two pieces of evidence, we can combine—or fuse—them using the so-called ‘Dempster– Shafer combination rule’, or ‘Dempster–Shafer fusion’ (DS fusion) which gives MZM 1 4M 2 defined by: MðAÞ Z ðM 1 4M 2 ÞðAÞ 8 1 X >< M Z 1KH 1 ðB 1 ÞM 2 ðB 2 Þ; for As:; ðB 1 ;B 2 Þ2U 2 =B 1 hB 2 ZA >: 0; for A Z : (2.1) Let us notice that the constant 2 3 H Z 1K X X 4 A3U M 1 ðB 1 ÞM 2 ðB 2 Þ5 Z X ðB 1 ;B 2 Þ2U 2 =B 1 hB 2 Z: ðB 1 ;B 2 Þ2U 2 =B 1 hB 2 ZAs: M 1 ðB 1 ÞM 2 ðB 2 Þ has an intuitive meaning and can be interpreted as the degree of conflict between the two pieces of evidence modeled by the bbas M 1 and M 2 . In the following, we will use the proportionality symbol ‘f‘ which is very practical to manipulate the DS fusion. Therefore, (2.1) will be written as: MðAÞ Z ðM 1 4M 2 ÞðAÞf X M 1 ðB 1 ÞM 2 ðB 2 Þ (2.2) B 1 hB 2 ZA knowing that As: and M(A)Z(M 1 4M 2 )(A) is obtained by dividing the r.h.s. of (2.1) by the sum 2 3 X 4 X M 1 ðB 1 ÞM 2 ðB 2 Þ5 A3U ðB 1 ;B 2 Þ3U 2 =B 1 hB 2 ZAs: As mentioned above, we will say that a bba M is ‘probabilistic’ when, being null outside singletons, it defines a probability and we will say that it is an ‘evidential’ bba when it is not probabilistic. As can be seen easily, when either M 1 or M 2 is probabilistic, the fusion result M is probabilistic. In particular, one can see that the classical calculus of the posterior probability is a Dempster–Shafer fusion (DS fusion) of two probabilistic bbas. For example, let us consider two random variables X and Y taking their values in UZ{u 1 ,u 2 } and R, respectively. Let M 0 be the law of X (which is a probability on U, and thus also a bba null outside singletons), and let p(yjxZu 1 ), p(yjxZu 2 ) be the distributions of Y conditional on XZu 1 and u 2 , respectively. For observed YZy, let M 1 the probability on U defined by M 1 ðu 1 ÞZpðyjxZu 1 Þ=ðpðyjxZu 1 ÞCpðyjxZu 2 ÞÞ and M 1 ðu 2 ÞZpðyjxZu 2 Þ=ðpðyjxZu 1 ÞCpðyjxZu 2 ÞÞ (which can be written M 1 (x)fp(yjx), where p(yjx) is the distribution of Y conditional on XZx). Then a very simple calculus shows that the posterior distribution of X, i.e. its distribution conditional on YZy, is the Dempster–Shafer fusion of M 0 with M 1 . This simple fact opens numerous perspectives of extension of the posterior
W. Pieczynski, D. Benboudjema / Image and Vision Computing 24 (2006) 61–69 63 distribution of X: if we replace either M 0 or M 1 by a bba which is not null outside singletons, then the fusion M 0 and M 1 gives a probability which is an extension of the posterior distribution of X. Such a fusion result can then be used, in a strictly same manner as the classical one, in different Bayesian techniques for estimating X from YZy. Studying these different extensions in the Markov fields context is the very aim of the paper. Let us consider the following two examples, which briefly describe how the DS fusion can be used in the image analysis context. Example 2.1. Let us consider the problem of satellite or airborne optical image segmentation into two classes UZ{u 1 , u 2 } ‘forest’ and ‘water’. Thus we have two random fields XZ (X s ) s2S and YZ(Y s ) s2S , each X s taking its values in UZ{u 1 , u 2 } and each Y s taking its values in R. Let us assume that there are clouds and let us denote by u c the ‘clouds’ class. Thus, in the classical probabilistic context, there are three conditional distributions on R:p(y s jx s Zu 1 ), p(y s jx s Zu 2 ) and p(y s jx s Zu c ). This classical model admits the following ‘evidential’ interpretation. If we are only interested on the classes UZ{u 1 ,u 2 }, the class u c brings no information about them, and thus u c models the ignorance and is assimilated to U. Finally, the classical probability q y s defined on {u 1 ,u 2 ,u c }by q y s ðu 1 Þfpðy s jx s Zu 1 Þ, q y s ðu 1 Þfpðy s jx s Zu 2 Þ, and q y s ðu 1 Þf pðy s jx s Zu c Þ is interpreted as a bba M 2 defined on P(U)Z{u 1 , u 2 ,U} byM 2 ({u 1 })fp(y s jx s Zu 1 ), M 2 ({u 2 })fp(y s jx s Zu 2 ), and M 2 (U)fp(y s jx s ZU). In other words, M 2 (U) models the ignorance attached with the fact that one cannot see through clouds. An interesting result states that when M 1 is a Markov probabilistic field distribution, its DS fusion with M 2 remains a Markov distribution, which generalizes the classical calculus of Markov posterior distribution [3]. Furthermore, when clouds disappear, M(U) becomes null, and such a ‘generalized’ Markov model becomes a classical HMF. So, the generalization we obtain embeds HMF particular case. Different Bayesian processing methods can then be based on the fused Markov distribution obtained this way. Roughly speaking, when the prior distribution is a Markov probabilistic bba, its fusion with a more general evidential bba does not pose problem and Bayesian processing remains workable in this more general context. Example 2.2. Let us consider the problem of segmentation of an observed bi-sensor image YZ(Y 1 ,Y 2 )Z(y 1 ,y 2 ) into three classes. So, we have UZ{u 1 ,u 2 ,u 3 }. Imagine that the first sensor Y 1 is sensitive to u 1 ,u 2 ,u 3 and {u 1 ,u 2 ,u 3 } (to fix ideas, it is an optical satellite sensor and there are clouds, modeled as in Example 2.1). Imagine that the second sensor Y 2 is sensitive to {u 1 ,u 2 } and u 3 (to fix ideas, it is an infrared satellite sensor and u 1 , u 2 are ‘hot’, while u 3 is ‘cold’). Let L 1 Z {{u 1 },{u 2 },{u 3 },{u 1 ,u 2 ,u 3 }}, L 2 Z{{u 1 ,u 2 },{u 3 }}, and L 1,2 Z{{u 1 },{u 2 },{u 3 },{u 1 ,u 2 }}. As above, y 1 defines a probability M 1 on L 1 and, in a similar way, y 2 defines a probability M 2 on L 2 .AsM 1 and M 2 are also bbas on P(U), we can fuse them, and the result M 3 ZM 1 4M 2 is a probability on L 1,2 . Finally, M 3 can be fused with a probabilistic Markov field, resulting in a Markov distribution which extends the classical posterior distribution. 3. Triplet Markov fields with evidential priors Let S be the set of pixels, and X,Y two random fields defined on S as specified in Section 1. Thus for each s2S, the variables X s and Y s take their values in UZ{u 1 ,.,u k } and R, respectively. The problem is to estimate X from Y, with immediate application to image segmentation. Considering a triplet Markov field (TMF), consists in introducing a third random field UZ(U s ) s2S , where each U s takes its values in LZ{l 1 ,.,l m }, and in assuming that TZ(X,U,Y) is a Markov field. The distribution of TZ(X,U,Y) is then a classical Gibbs one: " pðtÞ Z g exp K X # 4 c ðt c Þ (3.1) c2C Classically, (3.1) means that the distribution of T verifies p(t s jt r ,rss)Zp(t s jt r ,r2W s ), where (W s ) s2S is some neighborhood system (W s is the set of neighbors of s). C is the set of cliques associated with (W s ) s2S (a clique is either a singleton, or a set of mutually neighbors pixels). In this model, X and Y are interpreted as usual, as for U, it can have physical meaning or not. More precisely, there are at least three situations in which U models some reality: (i) the noise densities are unknown, and are approximated by Gaussian mixtures; (ii) there are subclasses in at least one class among u 1 ,.,u k ; and (iii) U models different stationarities of the field X (see [1] for (i) and (ii), and [2] for (iii)). Important is that (X,U) can be classically simulated according to the Markov distribution p(x,ujy), which enables to implement their different Bayesian estimations methods. For example, the classical Maximum Posterior Mode (MPM) method gives ^xZð^x s Þ s2S such that for each s2S pð^x s jyÞZmax xs 2U pðx s jyÞ. This estimate can be computed once the posterior marginal distributions p(x s jy) are known: as p(x s ,u s jy) can be classically estimated from a simulated sample, p(x s jy) is given by pðx s jyÞZ P u s 2L pðx s ; u s jyÞ. Of course, U can be estimated similarly if it is of interest [2]. We now deal with the main point of this paper, i.e. we want to explain how the classical computing of the posterior distribution in hidden Markov fields can be extended to DS fusion when the hidden data become ‘evidential’. Let us consider the classical hidden Markov field, with pðxÞZ g exp K P Q c2C 4 c ðx c Þ and pðyjxÞZ s2S pðy s jx s Þ. Then the distribution of (X,Y) is defined by " pðx; yÞ Z g exp K X 4 c ðx c Þ C X # Log½pðy s jx s ÞŠ (3.2) c2C s2S The key point is to use the observation that p(xjy) given by (3.2) can be seen as the result of the DS fusion of pðxÞZ g exp K P c2C 4 c ðx c Þ with the probability distribution q y (x) (y is fixed) obtained by normalizing p(yjx). More precisely, putting
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